<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E2:57_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E3:57_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:57_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:57_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="txt" href="wellord1.html#E2:57"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wellord1.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E4:57_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:57_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:57_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:57_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:57_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E7:57_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E2:57_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wellord1.html#E3:57_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#T57" target="_self">Th57</a></i>;<br/><b>hence </b><a NAME="E8:57_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E4:57_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="wellord1.html#E5:57_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="wellord1.html#T50" target="_self">Th50</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E2:57_2"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E3:57_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:57_2_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:57_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="txt" href="wellord1.html#E2:57"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wellord1.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E4:57_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:57_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:57_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:57_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:57_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E7:57_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E2:57_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E3:57_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="wellord1.html#E4:57_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wellord1.html#E5:57_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="wellord1.html#T57" target="_self">Th57</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:57"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E6:57"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E3:57"><i><font color="Green">E56</font></i></a>, <a class="txt" href="wellord1.html#E4:57"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div>
